An interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by G. Cantor in vol. 2 of the _Acta Mathematica_. Applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in J. A. Serret's _Cours d'Algebre Superieure_. 2. _Recurring Simple Continued Fractions._--The infinite continued fraction 1 1 1 1 1 1 1 1 1 1 a1 + -- -- --- -- -- --- -- -- --- -- a2 + a3 ... + a_n + b1 + b2 ... + b_n + b1 + b2 ... + b_n + b1 + ..., where, after the n^{th} partial quotient, the cycle of partial quotients b1, b2, ..., b_n recur in the same order, is the type of a recurring simple continued fraction. The value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. Since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. Conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. Thus __ 1 1 1 1 1 2 - \/ 3 = -- -- -- -- -- 3 + 1 + 2 + 1 + 2 + ..., ___ 1 1 1 1 1 1 1 1 \/ 28 = 5 + -- -- -- -- -- -- -- -- 3 + 2 + 3 + 10 + 3 + 2 + 3 + 10 + ... The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. There is only one non-recurring partial quotient a1. If b1, b2, ..., b_n is the cycle of recurring quotients, then b_n = 2a1, b1 = b_{n-1}, b2 = b_{n-2}, b3 = b_{n-3}, &c. In the case of a recurring continued fraction which represents [sqr]N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^2_{nr} -Nq^2_{nr} = (-1)^{nr}, whence, if n is odd, integral solutions of the indeterminate equation x squared - Ny squared = +-1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x squared -Ny squared = +1 can be found. The theory and development of the simple recurring continued fraction is due to Lagrange. For proofs of the theorems here stated and for applications to the more general indeterminate equation x squared -Ny squared = H the reader may consult Chrystal's _Algebra_ or Serret's _Cours d'Algebre Superieure_; he may also profitably consult a tract by T. Muir, _The Expression of a Quadratic Surd as a Continued Fraction_ (Glasgow, 1874). _The General Continued Fraction._